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# Fuzzy Logic

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1. Fuzzy Logic 11/6/2001

2. Agenda • General Definition • Applications • Formal Definitions • Operations • Rules • Fuzzy Air Conditioner • Controller Structure

3. General Definition Fuzzy Logic - 1965 Lotfi Zadeh, Berkely • superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth • central notion of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. • deals with real world vagueness

4. Applications • ABS Brakes • Expert Systems • Control Units • Bullet train between Tokyo and Osaka • Video Cameras • Automatic Transmissions

5. Formal Definitions • Definition 1: Let X be some set of objects, with elements noted as x. • X = {x}. • Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases. • Definition 3: A is EMPTY iff for all x, mA(x) = 0.0. • Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB]. • Definition 5: mA' = 1 - mA. • Definition 6: A is CONTAINED in B iff mA  mB. • Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)). • Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

6. Operations A B A  B A  B A

7. Controller Structure • Fuzzification • Scales and maps input variables to fuzzy sets • Inference Mechanism • Approximate reasoning • Deduces the control action • Defuzzification • Convert fuzzy output values to control signals

8. Rule Base • Fan Speed • Set stop {0, 0, 0} • Set slow {50, 30, 10} • Set medium {60, 50, 40} • Set fast {90, 70, 50} • Set blast {, 100, 80} Air Temperature • Set cold {50, 0, 0} • Set cool {65, 55, 45} • Set just right {70, 65, 60} • Set warm {85, 75, 65} • Set hot {, 90, 80}

9. Membership function is a curve of the degree of truth of a given input value default: The truth of any statement is a matter of degree Rules Air Conditioning Controller Example: • IF Cold then Stop • If Cool then Slow • If OK then Medium • If Warm then Fast • IF Hot then Blast

10. Fuzzy Air Conditioner

11. Mapping Inputs to Outputs